Multipath resolving correlation interferometer direction finding

ABSTRACT

Apparatus and a method utilizing correlation interferometer direction finding for determining the azimuth and elevation to an aircraft at long range and flying at low altitudes above water with a transmitting radar while resolving multipath signals. The signals from the radar are received both directly and reflected from the surface of the water using horizontally polarized and vertically polarized antenna arrays, are digitized and are stored in separate covariant matrices. Eigenvalues for the eigenvectors of the matrices processed on signal samples recorded on horizontally polarized X arrays are compared to the eigenvalues for the eigenvectors of the covariance matrices processed on signal samples recorded on vertically polarized X arrays. Incident field polarization is associated with the antenna array measurements that yield the strongest eigenvalue. The eigenvector and eigenvalues for the strongest signal are selected and used for subsequent signal processing. An initial global search assuming mirror sea-state reflection conditions using the signal eigenvector having the strongest eigenvalue is performed to yield an approximate elevation α and azimuth β to the aircraft. The approximate values are then used as the starting point for a subsequent conjugate gradient search to determine accurate elevation α and azimuth β to the aircraft.

FIELD OF THE INVENTION

The present invention relates to a correlation interferometer thatresolves radar signals received directly from a transmitter on anaircraft flying at low altitudes above the water and severe multi-pathradar signals from the same transmitter reflecting from the surface ofthe water to obtain the azimuth and elevation of the signal whileresolving multipath effects.

BACKGROUND OF THE INVENTION

A typical DF interferometer system locates the direction to a remotetransmitter by utilizing the phase difference of the transmitter signalarriving at the individual antennas. DF accuracy of such systems isdirectly related to DF array aperture size which is determined by thespacing between multiple antennas of an antenna array of the DF system.All other things being equal, larger DF apertures increase direction ofarrival (DOA) accuracy. However, simply increasing DF aperture sizeswithout increasing the number of DF antennas leads to large amplitudecorrelation side lobes and a real potential for large errors.

A basic problem has been to use such a prior art DF interferometersystem to locate the elevation and azimuth of a transmitter, such as aradar transmitter, mounted on an aircraft that is flying at lowaltitudes above water. Signals transmitted from the aircraft mountedradar received by line of sight and reflected from the surface of thewater introduce multi-path error into both azimuth and elevationmeasurements and degrade the reliability of the estimate of the azimuthand elevation of the directly incident radar signal. In the presence ofmulti-path effects, the wave fronts of the received signal are distortedso that the gradients of a wave front at a given location may be erraticand inconsistent with respect to the location of the signal source.

In addition, either a phase comparison interferometer or an amplitudecomparison direction-finding system over a reflecting surface such asseawater will result in multi-path caused errors in the measurement ofsignal direction-of-arrival. The occurrence of DOA errors due tomulti-path propagation is a function of the transmitter elevation angle,frequency, and the roughness of the surface of the seawater.

U.S. Pat. No. 5,568,394, issued Oct. 22, 1996 and entitled“Interferometry With Multipath Nulling” teaches a method that processesinterferometer data to provide for rejection of multi-path signalreturns from an emitter and computes an improved estimate of therelative angle between the emitter and an interferometer.

To do this interferometric data is gathered that comprises complexsignal amplitudes derived from the emitter at a plurality of emitterangles relative to the interferometer antenna array. The complex signalamplitudes derived at each of the plurality of emitter angles areprocessed by maximizing a predetermined log likelihood functioncorresponding to a natural logarithm of a predetermined probabilitydensity function at each of the plurality of emitter angles to produce aplurality of maximized log likelihood functions. The improved estimateof relative angle between the emitter and the interferometer is made byselecting the emitter angle corresponding to an optimally maximized loglikelihood function. The present method rejects multi path signalreturns from an emitter and computes an improved estimate of the anglebetween the emitter and the interferometer array. The processing methoduses a maximum likelihood function that incorporates multi-pathstatistics so that it is robust against multi-path variability. Thepresent processing method may also be employed to reject radomereflections in radars, particularly those employing antennas having arelatively low radar cross-section.

There are a number of shortcomings to the system taught in U.S. Pat. No.5,568,394 as compared to the present invention. The patent relates tonulling of multi-path reflections that are stable and repeatable, suchas own ship multi-path reflections, including signal blockage. Thistechnique uses previously recorded interferometer data that includesthese multi-path effects. It does not and cannot resolve multi-pathsignals, such as reflected from the surface of the ocean, but identifiesthe most probable incident wave arrival angle based on previousinterferometer array calibrations. There is no mention of polarizationdiverse antenna arrays and the effect of incident field polarization onthe multi-path nulling process.

U.S. Pat. No. 5,457,466, issued Oct. 10, 1995 and entitled “Emitterazimuth and elevation direction finding using only linear interferometerarrays” teaches a direction finding system for using a single linearinterferometer array mounted on a moving aircraft to make angle ofarrival (AOA) measurements only in sensor coordinates to perform emitterdirection finding. True azimuth and elevation to an emitter isdetermined.

Determining accurate angle-of-arrival (AOA) information for lowelevation targets using correlation interferometer direction finding isdescribed in a paper by K. A. Struckman, Resolution Of Low ElevationRadar Targets And Images Using A Shifted Array Correlation Technique,IEEE Antenna Propagation Society International Symposium (1989), pp.1736-1739, Jun. 1989, Vol. 3.

These linear interferometer arrays generates a direction of arrival(DOA) vector to provide azimuth with no coning error and elevation forlocation of an emitter. The elevation is derived from phase measurementsof signals received from the emitter in a way that allows sequentialaveraging to reduce azimuth and elevation range estimate errors.

The system generates virtual spatial arrays from the linear array basedon the aircraft's six degrees of freedom or motion. Six degrees offreedom refers to the six parameters required to specify the positionand orientation of a rigid body. The baselines at different times areassumed to generate AOA cones all having a common origin; theintersection of these cones gives the emitter DOA, from which azimuthelevation range can be derived. The generation and intersection of theAOA cones can be done in seconds, as opposed to the conventionalmulti-cone AOA approach, bearings only passive ranging, discussed above.Bearings only passive ranging requires that the origin of the cones beseparated by some intrinsic flight path length in order to form atriangle, and subtend bearing spread at the emitter.

There are a number of shortcomings to the azimuth—elevation DF systemtaught in U.S. Pat. No. 5,457,466 as compared to the present invention.The system is designed to operate on a moving aircraft. There is nomention of polarization diverse antenna arrays and the effect ofincident field polarization on the direction finding process. There isno mention of multi-path effects on the operation and accuracy of thesystem. Such multi-path effects are addressed and solved by theapplicant's invention. In addition, ambiguous baselines must beresolved.

Thus, there is a need in the prior art for improved DF systems that cancompensate for multi-path effects by rejecting the multi-path signalsand provide accurate azimuth and elevation measurements for an aircraftwith a transmitting radar flying at low altitudes above water.

SUMMARY OF THE INVENTION

The need in the prior art for an improved DF system that can compensatefor multi-path effects by rejecting the multi-path signals and providefor accurate azimuth and elevation measurements to an aircraft at longrange and flying at low altitudes above water with a transmitting radaris satisfied by the present invention. The improved system utilizescorrelation interferometer direction finding using an expanded, improvedversion of the technique of correlation that is described in a paper byN. Saucier and K. Struckman, Direction Finding Using CorrelationTechniques, IEEE Antenna Propagation Society International Symposium,pp. 260-263, Jun. 1975.

High multi-path content received signals are processed using acorrelation based signal processing algorithm that provides MultipathResolving Correlation Interferometer Direction Finding (MR-CIDF). Thenovel MR-CIDF processing implements a multi-path resolving correlationinterferometer that provides high precision direction-of-arrival (DOA)bearings in a severe multi-path environment. Stated another way theMR-CIDF provides a high precision estimate of the true arrival angle ofa signal of interest by correlation based processing that resolves boththe direct and reflected multi-path signals. This innovative approachdevelops robust and accurate DOA estimates by computing both thedirectly incident and multi-path components of signals contained withinthe signal of interest Field-of-View (FOV), which for this applicationis an azimuth span of ±45 degrees and an elevation span of 0° to 20°.

The field of view (FOV) of the DF array antenna is composed of tworegions, the positive alpha (+α) and negative alpha (−α) space. Thepositive alpha space is the region defined by an elevation=0° to +20°over the DF antenna array's azimuth coverage of ±45 degrees relative tothe broadside direction, β=0°. The negative alpha space is thejuxtaposed region mirrored about the horizon defined by an elevationextent of 0° to −20°.

The DF array comprise two cross (X) antenna arrays each having elevenlogarithmically spaced positions at each of which are located a pair ofantenna elements. The spacing of the antenna positions of one X array isdifferent than the spacing of the antenna positions of the other X arrayto provide coverage over two adjacent frequency bands. The two X arraysthereby operate over a combined bandwidth of 6:1. Each of the elevenantenna positions in both the X arrays has two antennas, one to receivevertically polarized signals and the other to receive horizontallypolarized signals. Both antennas at each position in each array arelinearly polarized Vivaldi notch (flared slot) antenna elements and theyare both connected to a fixed beam former. The beam former develops acosecant squared beam shape with the peak of the formed beam pointed atthe horizon. The slot of one of each pair of the Vivaldi notch antennasis oriented to receive horizontally polarized signals and the slot ofthe other of the Vivaldi notch antennas is oriented to receivevertically polarized signals.

The analog outputs of the eleven antenna element beam formers in each ofthe X arrays are connected to an antenna switch so that the elevenantenna elements of one X array or the other X array may be selected andprocessed. The selected signals pass through the antenna switch and areinput to a bank of receivers to be received and digitized as real andimaginary components (i and q) before signal samples are stored in acovariant matrix. In addition, the antenna switch functions to connectthe signals from the antenna elements receiving one type of polarizedsignals in an array to the receivers before connecting the signals fromthe antenna elements receiving the other type of polarized signals inthe same array. Thus, horizontally polarized signals are received andstored separately from the vertically polarized signals in thecovariance matrices.

Ideally numerous measurements or samples of a received signal ofinterest (SOI) are digitized and the data is inserted in real andimaginary format into a measured covariance matrix. Signal eigenvectorsare then computed from each covariant matrix. These eigenvectors areinserted into a ratio of quadratic forms correlation function that isused to compute the correlation between the signal eigenvectors andarray steering vectors retrieved from previously recorded arraycalibration manifolds. The ratio of quadratic forms function ismaximized over the reflected signal reflection coefficients.

The eigenvalues for the eigenvectors of the matrices generated by thesignal samples recorded on the horizontally polarized array are thencompared to the eigenvalues for the eigenvectors of the covariancematrices generated by the signal samples recorded on the verticallypolarized array to determine which signal polarization has the strongesteigenvalue. That eigenvector and the eigenvalues for that signal areselected and used for subsequent signal processing.

An initial global search assuming mirror sea-state reflection conditionsusing the signal eigenvector having the strongest eigenvalue withsteering vectors retrieved from the calibration array manifold is thenperformed upon making assumptions including that there are mirrorreflection conditions off the surface of the water. This search is not aconjugate gradient search. This initial search is a fairly coarse searchover all the data stored in the matrices to quickly find the highestpeak of the many peaks in the data and this yields first, approximatevalues for elevation α and azimuth β to the aircraft flying low overwater while emitting electromagnetic signals. The approximate values ofazimuth and elevation are then used as the starting point for a fineconjugate gradient search that uses the selected array manifold in theregion of the approximate values for elevation α and azimuth β toconverge to the exact values of azimuth and elevation to the aircraft inthe +α space. Being as the conjugate gradient search starts near thetrue peak in the stored data, as shown in a correlation surfacereflecting the data, all the data stored in the matrices need not besearched and this speeds the searching.

MR-CIDF DOA computations are based on the subsequent conjugate gradientmulti-dimensional search for the minimum of the functionF(1−R²)=1−MR-CIDF(|R|²) using the data stored in the selected covariantmatrices. The term (1−R²) causes the result of the conjugate gradientsearch to be a minimum rather than a maximum. The approximate values(α^(a), β^(a)) for the starting point in the neighborhood of the(1−|R|²) minimum is selected during the global search as described inthe previous paragraph. Next the gradient of F(1−R²) as a function ofthe upper hemisphere α and β and lower hemisphere −α+Δα, β+Δβ) values iscomputed. Next, a line search in the direction of this minimum in thisfirst gradient is computed. At this line search minimum, a new gradientis computed which is then inserted into a conjugate gradient searchroutine that computes the next direction for a new line search which isthen searched for a minimum. These search steps quickly converge to aglobal minimum, which is identified by slope gradient computations thatare approximately equal to zero. The correlation function MR-CIDF (|R|²)is always less than unity, therefore F(1−R²) is always greater thanzero. The conjugate gradient process minimizes F(1−R²) as a function ofupper DOA and lower reflection of arrival directions. The desired upperhemisphere DOA is identified as the direction to the low flyingaircraft.

To correct for array distortions, such as electromagnetic scatteringcaused by the platform on which the X arrays are mounted, a calibrationarray manifold correlation table constructed during system calibrationis used during processing. The calibration of the X arrays is developedin an accurately calibrated compact range. Array calibration data iscollected in one degree alpha and beta increments of the array's FOV.Array calibrations are stored in precise real and imaginary values forboth vertical and horizontal polarizations. Fine grained DOA processingdepends on the accurate interpolation of the array calibration manifold.

DESCRIPTION OF THE DRAWINGS

The invention will be better understood upon reading the followingDetailed Description in conjunction with the drawing in which:

FIG. 1 is a drawing showing radar signals received on a ship bothdirectly from an aircraft or missile and as multi-path reflections fromthe surface of water over which the aircraft or missile is flying at lowaltitude;

FIG. 2 is a schematic block diagram of a DF system that utilizes theteaching of the present invention to provide azimuth and elevationinformation for a radar transmitting from an aircraft or missile that isflying at low altitudes above water;

FIG. 3 shows a dual X array, logarithmic DF antenna utilized with theinvention;

FIG. 4 shows a correlation surface produced by graphing sampled dataassuming mirror image reflection off the surface of the water todetermine a starting point for a subsequent conjugate gradient basedcorrelation search to determine an accurate DOA to an aircraft ormissile flying low over water;

FIG. 5 is an equation used to process signal eigenvectors and eigenvalues and perform a conjugate gradient search to determine an accurateDOA to a low flying aircraft or missile;

FIG. 6 shows an expanded numerator of the equation shown in FIG. 5;

FIG. 7 shows an expanded denominator of the equation shown in FIG. 5;

FIG. 8 is a graph showing the results of a conjugate gradient searchingand the minimization of (1−|R|²) achieved thereby;

FIG. 9 shows two graphs of how the gradient search process achieves aconvergence to the correct DOA value; and

FIG. 10 shows five vertically polarized Vivaldi notch antennas connectedto a vertically polarized beam former and the ideal beam patternachieved thereby.

DETAILED DESCRIPTION

In the following detailed description and the drawings there arenumerous terms used that are defined below:

-   A_(pol) (α,β) are the calibration vectors as a function of α (the    elevation angle) and β (the azimuth angle) where:-   A_(pol)=A_(v)(vertical) or A_(h) (horizontal) calibration array    manifold data base.-   +α=the space above the surface of the water.-   −α=the space below the surface of the water.-   CIDF=Correlation Interferometer Direction Finding.-   MR-CIDF=Multipath Resolving Correlation Interferometer Direction    Finding-   DF=direction finding.-   DOA=direction of arrival, and consists of both the azimuth angle β    and elevation angle α of a received signal.-   E=electromagnetic radio waves incident on the array of antennas.-   Na=number of antennas in each beam forming/direction finding array,    eleven herein.-   (i, q)=in-phase and quadrature-phase of a complex quantity.-   Rxx=measured covariance matrix.-   λ=eigenvalues of the measured covariance matrix.-   Q=signal eigenvector of the measured covariance matrix.-   ρ_(d)=the direct complex coefficient-   ρ_(r)=the reflected complex coefficient-   SNR=signal-to-noise ratio.-   U_(pol)=is the eigenvector resolved received signal vector and is    composed of the direct component (+α space) and the reflected    component (−α space) where:-   U_(pol)=U_(v) (vertical receive array) or U_(h) (horizontal receive    array).-   α=elevation angle-   β=azimuth angle-   *=a complex conjugate

MR-CIDF correlation squared, |R(α,β)|², is defined by the equation inFIG. 5 where the terms in the equation are defined above. MR-CIDF ismaximized over the DOA parameters α, β and the complex coefficientsρ_(d), ρ_(r). Maximization of MR-CIDF over α, β is achieved by scanningover the array manifold data base. Maximization of MR-CIDF over ρ_(d),ρ_(r) is achieved by a observing that MR-CIDF is the ratio of quadraticforms that can be maximized in closed form. This maximization isdescribed as follows.

We expand the numerator of the equation in FIG. 5 which defines thesquared correlation term |R(α,β)|² to get the expanded numerator termshown in FIG. 6.

We expand the denominator of the equation in FIG. 5 which defines thesquared correlation term |R(α,β)|² to get the expanded denominator termshown in FIG. 7.

To simplify the description of the closed form maximization process, letH and G terms be defined as follows in Equation EQ 1:

$\begin{matrix}{{H_{11} = {{\sum\limits_{{na} = 1}^{N}\left\{ {{A\left( {\alpha,\beta,{na}} \right)}^{*} \times {U({na})}} \right\}}}^{2}}{H_{12} = {\sum\limits_{{na} = 1}^{N}{\left\{ {{A\left( {\alpha,\beta,{na}} \right)}^{*} \times {U({na})}} \right\} \times {\sum\limits_{{na} = 1}^{N}\left\{ {{A\left( {{{- \alpha} + {\Delta\alpha}},{\beta + {\Delta\beta}},{na}} \right)} \times {U({na})}^{*}} \right\}}}}}{H_{21} = {\sum\limits_{{na} = 1}^{N}{\left\{ {{A\left( {\alpha,\beta,{na}} \right)}^{*} \times {U({na})}} \right\}^{*} \times {\sum\limits_{{na} = 1}^{N}\left\{ {{A\left( {{{- \alpha} + {\Delta\alpha}},{\beta + {\Delta\;\beta}},{na}} \right)}^{*} \times {U({na})}} \right\}}}}}{H_{22} = {{\sum\limits_{{na} = 1}^{N}\left\{ {{A\left( {{{- \alpha} + {\Delta\alpha}},{\beta + {\Delta\beta}},{na}} \right)}^{*} \times {U({na})}} \right\}}}^{2}}{G_{11} = {\sum\limits_{{na} = 1}^{N}{{{U({na})}}^{2} \times {\sum\limits_{{na} = 1}^{N}{{A\left( {\alpha,\beta,{na}} \right)}}^{2}}}}}{G_{12} = {\sum\limits_{{na} = 1}^{N}{{{U({na})}}^{2} \times {\sum\limits_{{na} = 1}^{N}\left( {{A\left( {\alpha,\beta,{na}} \right)}^{*} \times {A\left( {{{- \alpha} + {\Delta\alpha}},{\beta + {\Delta\beta}},{na}} \right)}} \right)}}}}{G_{21} = G_{12}^{*}}{G_{22} = {\sum\limits_{{na} = 1}^{N}{{{U({na})}}^{2} \times {\sum\limits_{{na} = 1}^{N}{{A\left( {{{- \alpha} + {\Delta\alpha}},{\beta + {\Delta\beta}},{na}} \right)}}^{2}}}}}} & \left( {{EQ}\mspace{14mu} 1\text{:}} \right)\end{matrix}$

Substituting the G and H terms in Equation 1 into the equation in FIG. 5we get the following equation EQ 2:

$\begin{matrix}{{{R\left( {\alpha,\beta} \right)}}^{2} = \frac{\begin{matrix}{{{\rho_{d}}^{2} \times H_{11}} + {\rho_{d}^{*} \times \rho_{r} \times H_{12}} +} \\{{\rho_{d} \times \rho_{r}^{*} \times H_{21}} + {{\rho_{r}}^{2} \times H_{22}}}\end{matrix}}{\begin{matrix}{{{\rho_{d}}^{2} \times G_{11}} + {\rho_{d}^{*} \times \rho_{r} \times G_{12}} +} \\{{\rho_{d} \times \rho_{r}^{*} \times G_{21}} + {{\rho_{r}}^{2} \times G_{22}}}\end{matrix}}} & \left( {{EQ}\mspace{14mu} 2\text{:}} \right)\end{matrix}$

The characteristic equation of Hermitian forms is: [H−γG]=0. The rootsto this equation are called the characteristic values of the pencil. Thelargest root of this equation is the maximum of the ratio of Hermitianforms shown in EQ 3 as follows.

$\begin{matrix}{\gamma_{\max} = {{maximum}\mspace{14mu}{of}\mspace{11mu}{\left( \frac{H\left( {x,y} \right)}{G\left( {x,y} \right)} \right).}}} & \left( {{EQ}\mspace{14mu} 3\text{:}} \right)\end{matrix}$

|R(α,β)|² has this form, thus the maximum value of |R(α,β)|² at anglesα, β, −α+Δαand, β+Δβ is the solution to the determinant shown in EQ 4:

$\begin{matrix}\begin{bmatrix}{H_{11} - {\gamma_{\max} \times G_{11}}} & {H_{12} - {\gamma_{\max} \times G_{12}}} \\{H_{21} - {\gamma_{\max} \times G_{21}}} & {H_{22} - {\gamma_{\max} \times G_{22}}}\end{bmatrix} & \left( {{EQ}\mspace{14mu} 4\text{:}} \right)\end{matrix}$

This determinant equation is solved by the quadratic equation shown inEQ 5: (EQ 5:)a×γ _(max) ² +b×γ _(max) +c=0a=G ₁₁ ×G ₂₂ +|G ₁₂|²b=-(H ₁₁ ×G ₂₂ +H ₂₂ ×G ₁₁)+(H ₁₂ ×G ₂₁ +H ₂₁ ×G ₁₂)c=H ₁₁ ×H ₂₂ −H ₁₂ ×H ₂₁=0

The maximum value of |R(α,β)|² is therefore shown in EQ 6 following:

$\begin{matrix}{{\max\left( {{R\left( {\alpha,\beta} \right)}}^{2} \right)} = {\gamma_{\max} = {- \frac{b}{a}}}} & \left( {{EQ}\mspace{14mu} 6\text{:}} \right)\end{matrix}$The final search for the maximum of |R(α,β)|² is achieved by conjugategradient searching for the minimum of 1 −|R(α,β)|²

The +α space is that elevation or space above the surface of the waterand the −α space is that elevation or space below the surface of thewater where the reflected image of the received signal comes from.

In the following detailed description the term direction of arrival(DOA) is the term that is used and includes both the azimuth angle β andelevation angle α of a received signal.

In the following description eigenspace is defined as follows. If R_(xx)is an Na×Na square matrix and λ is an eigenvalue of R_(xx), then theunion of the zero vector 0 and the set of all eigenvectors correspondingto eigenvalues λ is known as the eigenspace of λ. The terms eigenvalueand eigenvector are well known in the art.

In the following description reference is made to eigenspacedecompositions. Eigenspace decompositions are well known in the art andare used in solving many signal processing problems, such as sourcelocation estimation, high-resolution frequency estimation, and beamforming. In each case, either the eigenvalue decomposition of acovariance matrix or the singular value decomposition of a data matrixis performed. For adaptive applications in a non-stationary environment,the eigenvalue decomposition is updated with the acquisition of new dataand the deletion of old data. For computational efficiency or forreal-time applications, an algorithm is used to update the eigenvaluedecomposition code without solving the eigenvalue decomposition problemfrom the start, i.e., an algorithm that makes use of the eigenvaluedecomposition of the original covariance matrix. In numerical linearalgebra, this problem is called the modified eigenvalue problem. In theexample of the invention disclosed herein, with only one signal beingreceived, the array vector for that signal is equal to its eigenvector.This is more fully described hereinafter.

In FIG. 1 is a drawing showing radar signals received on a ship 28 bothdirectly from an aircraft or missile 29 and as a multi-path reflectionfrom the surface of water 30 over which the aircraft 29 is flying at lowaltitudes. The multi-path reflection makes it appear to DF equipment onship 28 that there is an image aircraft or missile 31 that is not real.

In FIG. 2 is a detailed block diagram of a DF system that utilizes theteaching of the present invention to provide elevation and azimuthinformation for transmitters located on aircraft or missile 29 flyinglow over water 30 as shown in FIG. 1. The novel Multipath ResolvingCorrelation Interferometer Direction Finding (MR-CIDF) processingutilizes a conjugate gradient based search routine to implement amulti-path resolving correlation interferometer that provides highprecision direction-of-arrival (DOA) bearings in a severe multi-pathenvironment. Stated another way the MR-CIDF provides a high precisionestimate of the true arrival angle of a signal of interest bycorrelation based processing that resolves both the direct andmulti-path reflected signals.

DF antenna array 11 comprises two logarithmically spaced cross (X) beamformed antenna arrays 12 and 13 of antenna elements lying in a planethat is tilted back twenty degrees from the vertical. These antennaarrays are not shown in FIG. 1 but are shown in and described withreference to FIG. 3. In FIG. 3 is shown the dual X array 11, DF antennautilized with the invention. There are four primary objectives for usingthe X array design: (1) a robust design concept to reduce the probablyof “wild” DF bearings under multi-path conditions; (2) an accurate DFarray that operates over a six to one ratio, maximum operating frequencyto minimum operating frequency; and (3) an azimuth aperture to RFwavelength ratio that is greater or equal to 10 and (4) switchselectable orthogonally polarized (vertical and horizontal) antennas.

In FIG. 2, two incident electromagnetic signals E(t) 24 and E(t) 25impinge on the individual X array antennas 12 and 13 of array 11. SignalE(t) 24 is a radar signal received directly by line of sight from aradar transmitter mounted on an aircraft or missile flying at a lowaltitude above water. Signal E(t) 25 is the radar signal from the sameradar transmitter but it is reflected from the surface of the water asshown in FIG. 1. Thus, signals E(t) 24 and E(t) 25 are the same signalreceived at approximately the same azimuth but the angles of elevationare different because the signal E(t) 25 is reflected from the surfaceof the water. In addition, the angles of elevation and the phasedifferences of the received signal E(t) 25 will vary as represented bythe three arrows in FIG. 1 because the water is not a smooth surface.There are typically swells and waves on the surface of the water.

The reflection coefficient of seawater depends on the polarization ofthe transmitted E-field. For a vertically polarized wave at 10 GHz, themagnitude of the reflection coefficient is 0.8 at a grazing angle of onedegree and the magnitude is 0.15 at a grazing angle of ten degrees. Fora horizontally polarized wave at 10 GHz, the magnitude of the reflectioncoefficient is 0.995 at a grazing angle of one degree and thecoefficient is 0.95 at a grazing angle of ten degrees. The difference inreceived power for the two polarizations at a 10° grazing angle is 16dB. A result of the smaller vertical reflection coefficient is that theseverity and frequency of occurrence of multi-path caused directionfinding errors may be substantially reduced for vertical polarization.This observation has been verified with tests at 9.3 GHz and 10°elevation.

When the reflecting surface is smooth or slightly rough, reflectionsfrom the surface are specular and follow the laws of classical optics;they are coherent in phase and direction. Reflections from a roughsurface are termed diffuse and are reflected in random directions.Diffuse scattered energy reaching the receiver antenna has random phase.When diffuse reflections occur from water wave facets, the grazing anglefor reflection from a single facet will also be random and a function ofthe facet slope. As a result, the amplitude of diffuse scattering isalso random, and the peak variation is greatest for verticalpolarization. MR-CIDF processing assumes that the ensemble of diffusepaths merge into a single composite reflected signal.

Whether a surface may be considered smooth or rough depends on themagnitude of the diffuse energy received relative to the specular energyreceived. For direction finding systems, a ratio of 10 dB may be used asthe criteria for selecting the transition point in surface roughness.

The signals E(t) 24 and E(t) 25 impinge on the individual antennaelements of X beam formed antenna arrays 12 and 13 at different times asdetermined by their angle of incidence upon the plane of array 11 andthe spacing of the individual antenna elements.

X antenna arrays 12 and 13 are used to receive and process signals fromdifferent portions of the electromagnetic spectrum so their physicaldimensions are different. The signals received by only one of these twoX antenna arrays is processed at any one time. Accordingly, antennaswitch 14 is utilized. Antenna switches are well known in the art. Bothof the X array antennas 12 and 13 each have twenty-two beam formedarrays consisting of Vivaldi notch (flared slot) antennas, as brieflydescribed above and as shown in and described in detail hereinafter withreference to FIG. 3. Switch 14 is used to minimize the number ofreceivers 15 that receive and process the signals from the twenty twobeam formed arrays. Switch 14 is controlled by computer 17 to switch thereceivers 15 to either eleven beam formed, vertically polarized, X arrayantennas, or to eleven beam formed, x array antennas.

FIG. 3 shows the DF antenna array 11 that is comprised of twologarithmically spaced cross (X) beam formed antenna arrays 12 and 13 ofantenna elements 12 a-12 k and 13 a-13 k. The spacing of the antennaelements of one X array is different than the spacing of the antennaelements of the other X array to provide coverage over two adjacentfrequency bands. The two X arrays thereby operate over a combinedbandwidth of 6:1. X array 12 comprises eleven antenna positions in eachof which are located two linearly polarized beam formed arraysconsisting of Vivaldi notch (flared slot) antenna elements. This is atotal of twenty-two antenna beam formers. X array 13 comprises elevenantenna positions in each of which are located two linearly polarizedbeam formed arrays. There is also a total of twenty-two beam formedantenna arrays for X array 13. The slots of one of each set of theVivaldi notch antennas, for each beam former, is oriented to receivehorizontally polarized signals and the slots of the other set of theVivaldi notch antennas is oriented to receive vertically polarizedsignals.

More particularly, X array 12 comprises eleven antenna positions 12 a-12k. At each of these eleven positions there are two sets of beam formedVivaldi notch (flared slot) antennas. One beam former at each positionis oriented to receive horizontally polarized signals, and the otherbeam former at each position is oriented to receive vertically signals.Similarly, X array 13 comprises eleven antenna positions 13-13 and thereare two beam formers at each of these positions that function in thesame manner.

X array 12 has two arms. The first arm has positions 12 a, 12 b, 12 c,12 d, 12 e and 12 f, and the second arm has positions 12 g, 12 h, 12 c,12 i, 12 j and 12 k. This is a total of eleven positions associated withX array 12. X array 13 also has two arms. The first arm has positions 13a, 13 b, 13 c, 13 d, 13 e and 13 f, and the second arm has positions 13g, 13 h, 13 c, 13 i, 13 j and 13 k. This is a total of eleven positionsassociated with X array 13. As described above there are two beam formedarrays at each position. It should be noted that the two beam formers atcenter position 12 c are used in both arms of X array 12, and the twobeam formers at center position 13 c are used in both arms of X array13.

Returning to switch 14, it is operated in two combinations to obtain oneset of signal samples. The following order of switching is for exampleonly. First, the horizontally polarized antennas in array positions 12 athrough 12 k are connected through switch 14 to receivers 15 a-15 k.Second, the vertically polarized antennas in array positions 12 athrough 12 k are connected through switch 14 to receivers 15 a-15 k.Alternatively, but much more costly, a total of twenty two receiverscould be used and no vertical to horizontal switching would need to beintroduced.

The signals from the eleven antenna elements of the portion of theselected one of the X antenna arrays 12 or 13 that pass through switch14 are represented by voltage terms V(1,t) through V(11,t). The voltageterms V(1,t) through V(11,t) denote the complex analog waveform envelopethat is output from each of the 11 beam formed Vivaldi slot antennaarrays connected through switch 14 and is the only quantity that conveysinformation. Mathematically each signal represents a radar signal ofinterest and a noise component η(n,t) and is represented by EQ 7.

$\begin{matrix}{{V\left( {n,t} \right)} = {{{E_{d}\left( {n,t} \right)}{A_{pol}\left( {\alpha,\beta} \right)}} + {\sum\limits_{l}^{K}{{E_{k,r}\left( {n,t} \right)}{A_{pol}\left( {{{- \alpha} + {\Delta\alpha}_{k}},{\beta + {\Delta\;\beta_{k}}}} \right)}}} + {\eta\left( {n,t} \right)}}} & \left( {{EQ}\mspace{14mu} 7\text{:}} \right)\end{matrix}$Where: E_(d)(n,t)=direct incident field at beam formed array n.

E_(k,r)(n,t)=k_(th) sea state reflected incident files at beam formedarray n.

A_(pol)=beam formed array response for the k_(th) glint angle.

K=number of diffuse relections

η(n,t)=noise associated with the n_(th) receiver channel.

The voltages V(1,t) through V(11) are input to a respective one of theeleven receivers 15 a-15 k as shown. The signals output from each ofreceivers 15 a-15 k are in digitized format and are forwarded tocomputer 17. Computer 17 Nyquist samples the signals from the selectedeleven beam formers and stores the digitized samples in covariancematrices in a manner well known in the art.

More particularly, signals output from receivers 15 a-15 k in digitalformat are sampled, converted to real and imaginary values and stored ina plurality of covariance matrices in circuit 19. The typical sequenceis to Nyquist sample the received signals and record a set of samplesfor each of the eleven beam formers and connected through switch 14 atany moment in time. A number of sets of these signal samples aremeasured and processed into individual measured covariance matrices.

In block 20 in FIG. 2 each set of covariance matrices undergo eigenspacedecomposition to produce signal eigenvectors (U array vectors) andeigenvalues having azimuth and elevation information for the direct andreflected radar wave arrival information. The eigenvectors andeigenvalues are forwarded to computer 17 via path 21. The functionsperformed in blocks 19 and 20 are performed by a processor in a mannerknown in the art. The processes performed in blocks 19 and 20 may beperformed by computer 17 but the functions performed are shown asseparate blocks 19 and 20 to aid in understanding the invention.Covariance matrices and eigenspace decompositions are both known in theart and are used in solving many signal processing problems, such assource location estimation, high-resolution frequency estimation, anddigital beam forming. The parallel receiver channel architecture ofcovariance matrix processing is used to ensure that all correlationsurfaces are associated with a single remote transmitter and that themeasured data is not corrupted by co-channel RF interference. Multipleco-channel signals are identified by observing measured covariancematrix eigenvalues. Single signal conditions establish one strong signaleigenvalue and Na−1 noise eigenvalues when intercepted and received onNa RF channels, here Na=11.

For adaptive applications in the non-stationary environment of thepresent invention, the eigenvalue decomposition is updated with theacquisition of new data and the deletion of old data every fewmilliseconds. This occurs for each of the previously mentioned two setsof signal samples taken from the horizontal or vertical beam formers atpositions 12 a- 12 k or 13 a-13 k. The highest peak in each new set ofupdated data is close to the highest peak in the previous set of data soa new, coarse global search is not performed on each new set of updateddata. Instead α and β values from the previous conjugate gradient searchare used as the starting point for the conjugate gradient search on thenew set of updated data. For computational efficiency or for real-timeapplications, an algorithm is used to update the eigenvaluedecomposition code without solving the eigenvalue decomposition problemfrom the beginning again, i.e., an algorithm that makes use of theeigenvalue decomposition of the original covariance matrix. In numericallinear algebra, this problem is called the modified eigenvalue problem.

A typical processing sequence is as follows, the received signals(V(1,t)-V(11,t)) received on the individual antennas 12 a- 12 k or 13a-13 k of an antenna switch 14 selected one of arrays 12 or 13 arepassed through the antenna switch to a plurality of receivers 15 a-15 kwhere they are digitized and sampled at a Nyquist sampling rate togenerate 1024 samples for each of the eleven outputs of the horizontaland vertical polarization beam formers. The sets of digitized signalsamples are processed into two individual covariance matrices one foreach of the polarization dependent beam formers. One set of signalsamples is for the signals from the vertically polarized antennas andthe other set of signal samples is for the signals from the horizontallypolarized antennas. These covariance matrices undergo eigenspacedecomposition to produce two sets of eleven signal array U vectors andeigenvalues that contain the incident and reflected transmitter azimuth(β) and elevation (α) and other information for each of the samplingperiods. The preferred way to develop an array vector is to decompose acovariance matrix as a signal eigenvector having an eigenvalue, andassociate an array vector with the signal eigenvector. The U eigenvectorassociated with the strongest vertical polarization or horizontalpolarization signal eigenvalue computation is selected for direction ofarrival (DOA) processing.

Selection of vertical or horizontal polarization is determined in thefollowing manner. The signal is vertically polarized if [U_(v),U_(v)]>[U_(h), U_(h)] and is horizontally polarized if [U_(v) ,]<[U_(h), U_(h)]and [U_(pol), U_(pol)]=the inner product of U_(pol).Having selected the polarization of the received signal as defined abovethen A=A_(pol) and U=U_(pol).

To correct for array distortions caused by the platform on which theantenna arrays 12 and 13 are mounted, a calibration array manifoldcorrelation table constructed during system calibration is accessed toread out calibration data over the azimuth β, at +α and −α elevationsectors of interest. This calibration data is initially correlated by aninitial global search, by the maximized correlation function describedshown in FIG. 5 over the wave arrival sector of interest with Δα and Δβset equal to zero, and assuming mirror reflection conditions. βrepresents azimuth and α represents elevation. The result is a pointmarked W in FIG. 4 indicating approximate values of α and β which areclose to the real values of α and β. The approximate values of α and βfrom the initial global search are used in a subsequent conjugategradient search as the starting point that is close to actual point W toquickly locate the real values of α and β. Being as the conjugategradient search starts near the true, corrected peak in the stored data,as shown in a correlation surface reflecting the data, all the datastored in the matrices need not be searched and this speeds thesearching.

More particularly, FIG. 4 graphically shows, at a middle operatingfrequency, the result of this global search correlation process as acorrelation surface for incident and reflected wave arrival angles atβ=8°, α=8°, Δα=0.3° and, Δβ=−0.75° for incident of reflectioncoefficients, ρ_(d)=1.0 and ρ_(r)=1.0. The global search correlationsurface shown in FIG. 4 exhibits narrow peaks and low correlation sidelobes indicating accurate DOA solutions and robust array performance.The resultant DOA solution based on this specular mirror reflectionbased computation is slightly in error since this initial correlationsurface based solution assumes that Δα=0° and Δβ=0°. In FIG. 4 thiserror is shown as point W at the highest peak, but the position of thehighest peak is in error because of the assumption made.

The correct DOA parameters should yield Δ60 =0.3° and Δβ=−0.75°. Thesecorrect DOA parameters are computed by a subsequent conjugate gradientbased search routine on the array manifold used to graph the correlationsurface shown in FIG. 4 that starts with the approximate α, β valuescomputed as described above for the initial global search with Δα=0 andΔβ=0 and hill-climbs over these parameters to the maximum of thecorrelation process described by the equation shown in FIG. 5. This |R|²maximization is computed by minimizing 1−|R|². The convergence of 1−|R|²to a minimum for the numerical example described above is shown in FIG.8. Since the numerical experiment was based on a 20 dB SNR, the minimumis not equal to zero but is equal to approximately 0.005. Theconvergence to the correct DOA values is shown in FIG. 9. The mirrorreflection values shown for +α and +β search space starts with β≈7.7°,α≈8° and converges approximately to the true direct incident vales ofβ≈8°, α≈8°. The mirror reflection values shown minus α and plus β searchspace starts with β≈7.7°, α≈8° and converges approximately to thecorrect reflected values of β≈8°, α≈8°, Δβ≈−0.75°, Δα≈0.3°. The smallresidual errors are due to noise introduced at the 20 dB SNR level. Asthe conjugate gradient search routine progresses over several hundredsearch steps, as shown in FIG. 8, and gets closer to the true values ofα and β the value of |R|² approaches a maximum value. When this value isused in 1−|R|² the resulting value approaches zero as shown in FIG. 8.However, it will never reach zero due to noise. This is described ingreater detail hereinbelow.

Maximization starting with mirror reflection conditions reduces theconjugate gradient initial starting parameter problem. The reason forthis is as follows. The log-periodic array design limits the correlationsurface to one single highest peak. If a starting point for a conjugategradient searching sequence is chosen on the sloped side of this singlehighest peak, and is near the peak, the conjugate gradient searchingwill rapidly climb to the peak and in the process resolve α, β, Δα andΔβ. Optimization of the correlation process also involves thecomputation of the direct and reflected coefficients, these terms arehowever computed as output parameters but are hidden within the closedform maximization process of the equation shown in FIG. 5.

Stated another way, the initial global search is a fairly coarse searchover all the data stored in the matrices to quickly find the highestpeak of the many peaks in the data and this yields first, approximatevalues for elevation α and azimuth β to the aircraft. The approximatevalues of azimuth and elevation are then used as the starting point forthe conjugate gradient search that uses the selected array manifold inthe region of the approximate values for elevation α and azimuth β toconverge to the exact values of azimuth and elevation to the aircraft inthe +α space. Being as the conjugate gradient search starts near thetrue peak in the stored data, as shown in a correlation surfacereflecting the data, all the data stored in the matrices need not besearched and this speeds the searching for the true value of elevation αand azimuth β to the aircraft.

For adaptive applications in the non-stationary environment of thepresent invention, the eigenvalue decomposition is updated with theacquisition of new data and the deletion of old data every fewmilliseconds. This occurs for each of the previously mentioned two setsof signal samples taken from the horizontal or vertical beam formers atpositions 12 a- 12 k or 13 a-13 k. The highest peak in each new set ofupdated data is close to the highest peak in the previous set of data soa new, coarse global search is not performed on each new set of updateddata. Instead α and β values from the previous conjugate gradient searchare used as the starting point for the conjugate gradient search on thenew set of updated data. For computational efficiency or for real-timeapplications, an algorithm is used to update the eigenvaluedecomposition code without solving the eigenvalue decomposition problemfrom the beginning again, i.e., an algorithm that makes use of theeigenvalue decomposition of the original covariance matrix. In numericallinear algebra, this problem is called the modified eigenvalue problem.

Conjugate gradient searching is well known in the prior art. See a bookby W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling,“Numerical Recipes”, Cambridge University Press, Cambridge UK, 1986. Ithas been demonstrated that for certain types of functions, quadraticfunctions for example, the use of conjugate gradient directionprocessing allows convergence to a solution more quickly than thegradient direction. The standard conjugate gradient direction, whencalculated at a given point, takes into account the direction of theprevious step as well as the newly calculated gradient direction at thegiven point. If the direction of the step previously taken is designatedby a vector B and the newly calculated gradient direction at the givenpoint is designated as D, then the conjugate gradient direction at thegiven point is defined by the vector C in equation EQ 8 immediatelybelow.(EQ 8:) C _(conj) =D+hB

The conjugate gradient solutions used herein are based on numericalgradient derivatives of the equation shown equation EQ 6.

Returning to FIG. 3, there are three primary objectives for using the Xantenna array design: (1) a robust design concept to reduce the probablyof “wild” DF bearings under multi-path conditions; (2) an accurate DFarray that operates over a six to one ratio, maximum operating frequencyto minimum operating frequency; and (3) an azimuth aperture to RFwavelength ratio that is greater or equal to 10.

The first objective is fulfilled by the fundamental X array design.First, there is a log periodic spacing between the antenna elements 12 athrough 12 i and 13 a-13 i to minimize the effects of the array to “wildDF bearings”. Second, the width dimension of the array is wider than theheight dimension to yield better azimuth accuracy than elevationaccuracy. Directly incident and sea surface reflected rays generate aninterferometer pattern across the face of the antenna array. Thisinterferometer pattern has horizontal symmetry and nulls will beobserved as horizontal lines across the total face of the array.Conventional L shaped antenna arrays based on horizontal and verticalrows of antennas arrays are very susceptible to “wild DF bearings”. Ifone antenna of the horizontal array is within a null, the wholehorizontal array will be nulled, creating erroneous measurements andattendant DF errors. By contrast, X arrays have only a single pair ofantennas exposed to common null regions. The other antennas of the Xarray continue to intercept strong signals and provide for accuratemeasurements, azimuth and elevation DF solutions.

The second objective is fulfilled by the two-band array design, eachsub-band covering a maximum to minimum ratio of 3:1. The two-band arraydesign achieves a total 6:1 maximum operating frequency to minimumoperating frequency bandwidth.

The third objective is solved by the double X array design shown in FIG.3. When the operating frequency is the geometric mean=square-root(maximum×minimum frequencies), each of these arrays has an azimuthaperture to wavelength ratio (Da/λ) equal to twenty. The elevationapertures are one third the dimension of the azimuth angle apertures. Atthe geometric mean operating frequency, the azimuth 3 dB beam widthequals 2.4 degrees, while the elevation 3 dB beam width equals 7.0degrees. DF solutions based on these large apertures will only be robustif the eleven element arrays are carefully designed as shown in FIG. 3.The eleven beam formed antennas of each array (12 a-12 k and 13 a-13 k)each have two diagonal rows of six beam formed elements that share acommon central antenna element (12 c, and 13 c). Two orthogonallyoriented antenna elements beam formers are installed at each of theeleven locations of each X array, one beam former designed for thereception of vertically polarized signals and the other beam former forthe reception of horizontal polarized signals. Switching with antennaswitch 14 is used to select the polarization that most nearly matchesthe incident electric field E(t) polarization. The six antennallocations along each leg of both the X-arrays 12 and 13 are chosen byusing a logarithmic spacing that minimizes interferometer side-lobes.Broadside DF accuracy (standard-deviation) under zero scattered fieldconditions and a 20 dB Signal-to-Noise ratio is: azimuth=0.0346 degrees,elevation=0.1144 degrees.

Each of the antenna elements of X array 12 and 13 are two sets oflinearly polarized Vivaldi notch antennas connected to fixed beamformers. FIG. 10 shows five vertically polarized Vivaldi notch antennasconnected to the vertically polarized beam former. This beam former isdesigned to generate cosecant squared beam shape with the peak of theformed beam pointed at the horizon. Horizontal Vivaldi antennas areinstallation to form horizontally polarized signal beam formers. Thevertical and horizontal beam formers are computer selected during themeasurement process. Ideally numerous measurements on the signal ofinterest (SOI) are recorded and inserted into the measured covariancematrix. The time duration of this process is limited by the amount oftime needed for the target to move from beyond the specified DFaccuracy. There is however a technique for summing correlation surfacesthat reflect target movement during the measurement process, such asdescribed in U.S. patent application Ser. No. 11/249,922, filed Oct. 13,2005 and entitled “Moving Transmitter Correlation InterferometerGeolocation”.

The Vivaldi notch antennas are a traveling wave antenna havingexponentially tapered notches which open outwardly from a feed at thethroat of the notch. Typically, in such a Vivaldi notch antenna there isa cavity behind the feed point which prevents energy from flowing backaway from the feed point to the back end of the Vivaldi notch. As aresult, in these antennas, one obtains radiation in the forwarddirection and obtains a single lobe beam over a wide frequency range.One can obtain a VSWR less than 3:1 with the beams staying fairlyconstant over the entire antenna bandwidth with the lobe having about aneighty degree or ninety degree beam width. The Vivaldi notch antennasare single lobe antennas having wide bandwidth and are unidirectional inthat the beam remains relatively constant as a single lobe over a 6:1bandwidth whether in elevation and in azimuth.

Vivaldi notch antennas were first described in a monograph entitled TheVivaldi Aerial by P. G. Gibson of the Phillips Research Laboratories,Redhill, Surrey, England in 1978 and by Ramakrishna Janaswamy and DanielH. Schaubert in IEEE Transactions on Antennas and Propagation, vol.AP-35, no. 1, September 1987. The above article describes the Vivaldiaerial as a new member of the class of aperiodic continuously scaledantenna structures which has a theoretically unlimited instantaneousfrequency bandwidth. This antenna was said to have significant gain andlinear polarization that can be made to conform to constant gain versusfrequency performance. One reported Gibson design had been made withapproximately 10 dB gain and a minus −20 dB side lobe level over aninstantaneous frequency bandwidth extending from below 2 GHz to about 40GHz.

One Vivaldi notch antenna is described in U.S. Pat. No. 4,853,704 issuedAug. 1, 1989 to Leopold J. Diaz, Daniel B. McKenna, and Todd A. Pett.The Vivaldi notch has been utilized in micro strip antennas for sometime and is utilized primarily in the high end of the electromagneticspectrum as a wide bandwidth antenna element.

While what has been described herein is the preferred embodiment of theinvention it will be appreciated by those skilled in the art thatnumerous changes may be made without departing from the spirit and scopeof the invention.

1. A method comprising determining the azimuth and elevation to anaircraft at long range and transmitting an electromagnetic signal whileflying at a low altitude above water, wherein the signal is receivedusing a plurality of vertically polarized antennas and horizontallypolarized antennas in an antenna array, wherein the received signal isreceived directly from the aircraft and is received via multi-pathreflections from the surface of the water, wherein the azimuth andelevation to the aircraft are defined by a conjugate gradient basedcorrelation search of covariance matrices containing the signalsreceived by the vertically polarized antennas and the horizontallypolarized antennas in digitized form representing the real and imaginarycomponents of the received signal, and wherein the following equation isused to process the digitized information stored in the covariancematrices is searched to find the azimuth and elevation to the aircraftusing the equation:${{R\left( {\alpha,\beta} \right)}}^{2} = \frac{\sum\limits_{{na} = 1}^{N}{{\begin{Bmatrix}{{\rho_{d} \times {A_{pol}\left( {\alpha,\beta,{na}} \right)}} + {\rho_{r} \times}} \\{A_{pol}\left( {{{- \alpha} + {\Delta\alpha}},{\beta + {\Delta\beta}},{na}} \right)}\end{Bmatrix}^{*} \times {U_{pol}({na})}}}^{2}}{\sum\limits_{{na} = 1}^{N}{{\begin{Bmatrix}{{\rho_{d} \times {A_{pol}\left( {\alpha,\beta,{na}} \right)}} + {\rho_{r} \times}} \\{A_{pol}\left( {{{- \alpha} + {\Delta\alpha}},{\beta + {\Delta\beta}},{na}} \right)}\end{Bmatrix}}^{2} \times {\sum\limits_{{na} = 1}^{N}{{U_{pol}({na})}}^{2}}}}$where |R(α,β)|² is the correlation squared and the maximum value issearched for during the conjugate gradient based correlation search, αis the elevation angle to the aircraft transmitting the signal, β is theazimuth angle to the aircraft transmitting the signal, ρ_(d) is thedirect complex coefficient, ρ_(r) is the reflected complexcoefficient, * is a complex conjugate, na is the number of antennas inthe beam forming array utilized to receive the signals transmitted bythe aircraft, A_(pol) are calibration vectors as a function of α and β,and U_(pol) is the eigenvector resolved received signal vector and iscomposed of the directly received signal component and the reflectedsignal component and is equal the sum of the eigenvectors of the signalsfrom the vertically polarized antennas and the horizontally polarizedantennas.
 2. The method for determining the azimuth and elevation to anaircraft of claim 1 wherein the signals received using the plurality ofvertically polarized antennas are digitized and stored in a firstcovariant matrix, and the signals received using the plurality ofhorizontally polarized antennas are digitized and stored in a secondcovariant matrix; wherein the first covariant matrix undergoeseigenspace decomposition to produce a first set of signal arrayeigenvectors U having first eigenvalues; wherein the second covariantmatrix undergoes eigenspace decomposition to produce a second set ofsignal array eigenvectors U having second eigenvalues; wherein the firsteigenvalues are compared to the second eigenvalues and the receivedelectromagnetic signal is vertically polarized if the first eigenvaluesare greater than the second eigenvalues, and is horizontally polarizedif the first eigenvalues are less than the second eigenvalues; andwherein the eigenvector with the associated highest eigenvalues are usedto perform the conjugate gradient based correlation search.
 3. Themethod for determining the azimuth and elevation to an aircraft of claim2 wherein before a conjugate gradient based correlation search isperformed using said equation to determine the azimuth and elevation tothe aircraft, a global correlation search is performed using saidequation but the terms Δα and Δβ in the equation are set equal to zeroand it is assumed that mirror reflection conditions for reflection ofthe electromagnetic signal from the surface of the water, the resultobtained using said equation are approximate values for elevation α andazimuth β to the aircraft, the approximate values then being used as thestarting point in the conjugate gradient based correlation search usingsaid equation.
 4. The method for determining the azimuth and elevationto an aircraft of claim 3 wherein the maximization of the term |R(α,β)|²over ρ_(d), ρ_(r) is achieved by a observing that |R(α,β)|² can be theratio of quadratic forms in closed form when the numerator of saidequation is defined as equal to${{\rho_{d}}^{2} \times {{\sum\limits_{{na} = 1}^{N}\left\{ {{A\left( {\alpha,\beta,{na}} \right)}^{*} \times {U(\;{na})}} \right\}}}^{2}} + {\rho_{d}^{*} \times \rho_{r} \times {\sum\limits_{{na} = 1}^{N}{\left\{ {{A\left( {\alpha,\beta,{na}} \right)}^{*} \times {U({na})}} \right\} \times {\sum\limits_{{na} = 1}^{N}\left\{ {{A\left( {{{- \alpha} + {\Delta\alpha}},{\beta + {\Delta\beta}},{na}} \right)}^{*} \times {U({na})}} \right\}^{*}}}}} + {\rho_{d} \times \rho_{r}^{*} \times {\sum\limits_{{na} = 1}^{N}{\left\{ {{A\left( {\alpha,\beta,{na}} \right)}^{*} \times {U({na})}} \right\}^{*} \times {\sum\limits_{{na} = 1}^{N}\left\{ {{A\left( {{{- \alpha} + {\Delta\alpha}},{\beta + {\Delta\beta}},{na}} \right)}^{*} \times {U({na})}} \right\}}}}} + {{\rho_{r}}^{2} \times {{\sum\limits_{{na} = 1}^{N}\left\{ {{A\left( {{{- \alpha} + {\Delta\alpha}},{\beta + {\Delta\beta}},{na}} \right)}^{*} \times {U({na})}} \right\}}}^{2}}$and the denominator of said equation is defined as equal to$\sum\limits_{{na} = 1}^{N}{{{U({na})}}^{2} \times \left\{ {{{\rho_{d}}^{2} \times {\sum\limits_{{na} = 1}^{N}{{A\left( {\alpha,\beta,{na}} \right)}}^{2}}} + {\rho_{d}^{*} \times \rho_{r} \times {\sum\limits_{{na} = 1}^{N}\left( {{A\left( {\alpha,\beta,{na}} \right)}^{*} \times {A\left( {{{- \alpha} + {\Delta\alpha}},{\beta + {\Delta\beta}},{na}} \right)}} \right)}} + {\rho_{d} \times \rho_{r}^{*} \times {\sum\limits_{{na} = 1}^{N}\left( {{A\left( {\alpha,\beta,{na}} \right)} \times {A\left( {{{- \alpha} + {\Delta\alpha}},{\beta + {\Delta\beta}},{na}} \right)}^{*}} \right)}} + {{\rho_{r}}^{2} \times {\sum\limits_{{na} = 1}^{N}{{A\left( {{{- \alpha} + {\Delta\alpha}},{\beta + {\Delta\beta}},{na}} \right)}}^{2}}}} \right\}}$and the closed form maximization process is further simplified byletting H and G terms be defined as$H_{11} = {{\sum\limits_{{na} = 1}^{N}\left\{ {{A\left( {\alpha,\beta,{na}} \right)}^{*} \times {U({na})}} \right\}}}^{2}$$H_{12} = {\sum\limits_{{na} = 1}^{N}{\left\{ {{A\left( {\alpha,\beta,{na}} \right)}^{*} \times {U({na})}} \right\} \times {\sum\limits_{{na} = 1}^{N}\left\{ {{A\left( {{{- \alpha} + {\Delta\alpha}},{\beta + {\Delta\beta}},{na}} \right)} \times {U({na})}^{*}} \right\}}}}$$H_{21} = {\sum\limits_{{na} = 1}^{N}{\left\{ {{A\left( {\alpha,\beta,{na}} \right)}^{*} \times {U({na})}} \right\}^{*} \times {\sum\limits_{{na} = 1}^{N}\left\{ {{A\left( {{{- \alpha} + {\Delta\alpha}},{\beta + {\Delta\;\beta}},{na}} \right)}^{*} \times {U({na})}} \right\}}}}$$H_{22} = {{\sum\limits_{{na} = 1}^{N}\left\{ {{A\left( {{{- \alpha} + {\Delta\alpha}},{\beta + {\Delta\beta}},{na}} \right)}^{*} \times {U({na})}} \right\}}}^{2}$$G_{11} = {\sum\limits_{{na} = 1}^{N}{{{U({na})}}^{2} \times {\sum\limits_{{na} = 1}^{N}{{A\left( {\alpha,\beta,{na}} \right)}}^{2}}}}$$G_{12} = {\sum\limits_{{na} = 1}^{N}{{{U({na})}}^{2} \times {\sum\limits_{{na} = 1}^{N}\left( {{A\left( {\alpha,\beta,{na}} \right)}^{*} \times {A\left( {{{- \alpha} + {\Delta\alpha}},{\beta + {\Delta\beta}},{na}} \right)}} \right)}}}$G₂₁ = G₁₂^(*)${G_{22} = {\sum\limits_{{na} = 1}^{N}{{{U({na})}}^{2} \times {\sum\limits_{{na} = 1}^{N}{{A\left( {{{- \alpha} + {\Delta\alpha}},{\beta + {\Delta\beta}},{na}} \right)}}^{2}}}}},$then the G and H terms are inserted into said equation to get a revisedequation${{R\left( {\alpha,\beta} \right)}}^{2} = \frac{\begin{matrix}{{{\rho_{d}}^{2} \times H_{11}} + {\rho_{d}^{*} \times \rho_{r} \times H_{12}} +} \\{{\rho_{d} \times \rho_{r}^{*} \times H_{21}} + {{\rho_{r}}^{2} \times H_{22}}}\end{matrix}}{\begin{matrix}{{{\rho_{d}}^{2} \times G_{11}} + {\rho_{d}^{*} \times \rho_{r} \times G_{12}} +} \\{{\rho_{d} \times \rho_{r}^{*} \times G_{21}} + {{\rho_{r}}^{2} \times G_{22}}}\end{matrix}}$ where the characteristic equation of Hermitian forms[H−γG]=0 and the largest root of this equation is the maximum of theratio of Hermitian forms is${\gamma_{\max} = {{maximum}\mspace{14mu}{of}\mspace{11mu}\left( \frac{H\left( {x,y} \right)}{G\left( {x,y} \right)} \right)}};$where |R(α,β)|² has this form and the maximum value of |R(α,β)|² atangles α, β, −α+Δα, and, β+Δβ is the solution to the determinantequation $\quad\begin{bmatrix}{H_{11} - {\gamma_{\max} \times G_{11}}} & {H_{12} - {\gamma_{\max} \times G_{12}}} \\{H_{21} - {\gamma_{\max} \times G_{21}}} & {H_{22} - {\gamma_{\max} \times G_{22}}}\end{bmatrix}$ where this determinant equation is solved by thequadratic equationα×γ _(max) ² +b×γ _(max) +c=0α=G ₁₁ ×G ₂₂ +|G ₁₂|²b=−(H ₁₁ ×G ₂₂ +H ₂₂ ×G ₁₁)+(H ₁₂ ×G ₂₁ +H ₂₁ ×G ₁₂)c=H ₁₁ ×H ₂₂ −H ₁₂ ×H ₂₁=0 and the maximum value of |R(α,β)|² istherefore${\max\left( {{R\left( {\alpha,\beta} \right)}}^{2} \right)} = {\gamma_{\max} = {- {\frac{b}{a}.}}}$